Trend to equilibrium for run and tumble equations with non-uniform tumbling kernels

TL;DR

This paper analyzes the long-term behavior of a bacterial run and tumble model with non-uniform tumbling kernels, proving convergence to equilibrium with explicit rates for different kernel types, a novel contribution in this area.

Contribution

It provides the first rigorous analysis of the long-time behavior of run and tumble equations with non-uniform tumbling kernels, including explicit convergence rates.

Findings

Convergence to equilibrium with exponential rate for angle-dependent kernels.

Algebraic convergence rate for Maxwellian velocity distribution.

First results on long-time behavior with non-uniform kernels.

Abstract

We study the long-time behaviour of a run and tumble model which is a kinetic-transport equation describing bacterial movement under the effect of a chemical stimulus. The experiments suggest that the non-uniform tumbling kernels are physically relevant ones as opposed to the uniform tumbling kernel which is widely considered in the literature to reduce the complexity of the mathematical analysis. We consider two cases: (i) the tumbling kernel depends on the angle between pre- and post-tumbling velocities, (ii) the velocity space is unbounded and the post-tumbling velocities follow the Maxwellian velocity distribution. We prove that the probability density distribution of bacteria converges to an equilibrium distribution with explicit (exponential for (i) and algebraic for (ii)) convergence rates, for any probability measure initial data. To the best of our knowledge, our results are the first results concerning the long-time behaviour of run and tumble equations with non-uniform tumbling kernels.